Weak fiber products in a bicategory of fractions

We fix any pair $(\mathcal{C},\mathbf{W})$ consisting of a bicategory and a class of morphisms in it, admitting a bicalculus of fractions, i.e. a “localization” of $\mathcal{C}$ with respect to the class $\mathbf{W}$. In the resulting bicategory of fractions, we identify necessary and sufficient conditions for the existence of weak fiber products.